It is frequently desirable to determine the characteristics of a spatial phase profile of a wavefront. In optics, this phase (divided by 2) is referred to as an Optical Path Difference (OPD) map, or aberration function. A knowledge of the OPD is essential to determine how the wavefront will propagate or how well the wavefront can be imaged. However, in that phase cannot be measured directly, a problem of phase retrieval is presented.
A conventional phase retrieval method employs interferometry, in which two coherent wavefronts are brought together (interfered) and the resulting high frequency intensity variations, referred to as fringes, are analyzed to determine the phase. A similar interference effect occurs in propagation and imaging. Interference effects in images also produce fringes which are referred to as diffraction fringes or sidelobes. Several of the better known properties of interferometry include the following. First, only phase differences can be determined and not the absolute phase. Second, for a single static interferogram, a sign ambiguity exists.
Some lesser known, but equally important properties of interferometry include the following. First, only intensities are measured. Phase cannot be measured directly, which is a result of a fundamental tenet of quantum mechanics regarding "observables". The phase is computed from intensity patterns or fringes which are recorded on film or by optical detectors. Second, and as a result of the immediately preceding property, it is an implicit assumption of interferometry that the amplitude of the two beams is constant, or at least slowly varying compared to the fringes, or is explicitly known so that the phase can be computed. This is required so that intensity variations, due to phase, can be differentiated from intensity variations due to amplitude. Similar constraints occur in microwave and FM systems. Third, adequate sampling of the fringes requires at least two samples per cycle.
An overview is now presented of the basic types of interferometers that provide phase retrieval from images.
An early type of interferometer is the Michaelson, in which an input beam is split into two parts, one of which is a reference and the other of which samples a surface under test. The two beams are recombined to form an interferogram which allows the phase difference between the surface under test and the reference beam to be computed. Multiple beam interferometers may be used instead of a Michaelson-type to obtain sharper fringes. However, in all cases the general approach is the same: the beam (or beams) from the surface under test is compared to a reference beam (or beams) and the phase is inferred from the resulting fringes.
A problem that arises during the use of a Michaelson (or similar interferometer) is a requirement for high stability between the two optical paths and a requirement that no unknown errors exist in the interferometer optics or reference.
Another common type of interferometer is the shearing interferometer, wherein the reference is the surface under test which is shifted in position. The shift may be generated by a shear plate or by other means. A significant difference between the shearing interferometer and the Michaelson interferometer is that the surface is compared with itself (instead of a reference) and spatial phase differences are obtained. For small shifts, the phase shifts from a shearing interferometer are summed to reconstruct the original wavefront phase. This type of interferometer is less sensitive to mechanical stability than is the Michaelson, but places more constraints on the quality and coherence of the source beam.
Although not usually referred to as interferometry, another form of interferometry is imaging or, more generally, propagation. Propagation processes are governed by the laws of diffraction. Although diffraction and interferometry and not generally associated, it has been found that diffraction results from the interference of light "scattered" from the edges of an aperture with the light transmitted through the aperture. The resulting interference fringes, usually called diffraction sidelobes or fringes, are a function of the aperture function, such as size and shape, and the distance of the fringes from the aperture as well as aberration in the aperture. In propagating a wavefront, light from all parts of the aperture is summed together (interfered) to calculate the propagated wavefront. This is similar to a shearing interferometer in which the wavefront is interfered with itself, as displaced. Under certain circumstances the phase in the aperture can be estimated from a measurement of an image.
Conventional wavefront phase retrieval methods from propagation or imaging are iterative, successive approximation approaches. These approaches are generally inaccurate and may not necessarily converge to a correct solution due to the existence of secondary maxima. When these conventional methods do converge, convergence is frequently very slow. In interferometry, significant amounts of hardware may be required to accomplish wavefront phase retrieval. The provision of additional hardware is especially disadvantageous in airborne and spaceborne applications, where size and weight are important factors.
It is thus an object of the invention to employ a measurement of an image to determine aberrations, thus eliminating a need for the hardware associated with interferometric apparatus and the stability requirements of maintaining optical alignments to a fraction of a wavelength.
It is a further object of the invention to provide method and apparatus for accomplishing Spatial Wavefront Evaluation by Intensity Relationships (SWEBIR).